Mathematical Models of Classical Electrodynamics · Tensor Approach; • III. Geometric Algebra...
Transcript of Mathematical Models of Classical Electrodynamics · Tensor Approach; • III. Geometric Algebra...
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Mathematical Models of Classical Electrodynamics
Sava V. Savov Department of Electrical Engineering
Technical University of Varna
Bulgaria
Geometry, Integrability and Quantization Varna, Bulgaria
June 2017
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Outline
• I. INTRODUCTION
• The Original Maxwell’s Equations
• II. DIFFERENT ELECTROMAGNETIC MODELS 1. The Concept of Vector Fields
2. The Tensor Notations
3. The Geometric Algebra Model
4. The Differential Forms Model III. CONCLUSION
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I. INTRODUCTION
• The Original Maxwell’s Equations
• In 1865 the great Scottish physicist James C. Maxwell published his spectacular theory that became a basis of Classical Electrodynamics (CED);
• They covered four different physical laws: 1) Faraday’s law; 2) Ampere’s law (with a correction introduced pure theoretically by Maxwell); 3) Gauss’s law of electric field; 4) Gauss’s law of magnetic field (no magnetic charges!).
• Originally: 20 equations, then: 12 equations. Ether included?
• Different EM Models
• I. The Concept of Vector Fields
• The vector notations (in 3-D space) are listed below. It is curious than even after introduction into CED, the main equations are remained the same in principle.
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I. The Concept of Vector Fields
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. I. The Concept’s of Vector Fields
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II. The Tensor Notations
• In 1905 Einstein – Minkowski (Special Theory of Relativity).
• 4 – D space-time frame of inertial coordinate systems:
(4)
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II. The Tensor Notations
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II. The Tensor Notations
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III. The Geometric Algebra Model
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III. The geometric algebra model
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III. The geometric algebra model
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IV. The Differential Forms Model
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IV. The Differential Forms Model
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IV. The Differential Forms Model
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IV. The Differential Forms Model
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IV. The Differential Forms Model
• The DF of Maxwell’s equations is:
• (18)
• The operator (d) transforms the forms as follows: 0-form -> 1-form -> 2-form -> 3-form -> 0-form. The Integral Form of Maxwell’s equations is:
• (19)
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IV. The Differential Forms Model
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IV. The Differential Forms Model
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III. CONCLUSION
• In this paper four different models of the Maxwell’s Equations (explained the Classical Electrodynamics) are considered and compared:
• I. Vector Approach;
• II. Tensor Approach;
• III. Geometric Algebra Model;
• IV. Differential Forms Model.
• The first one is a classical approach (19th century), introduced by Heaviside. The other ones are modern: they are introduced in the beginning of 20th century. All they have one important advantage: the original 8 scalar equations are converted to only 2 ones.
• This brief notation allows not only to deal with short system, but easy to check the important property called “covariance”.
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• Thank you for your attention!
• Any questions?